Law of Reciprocal

Q.

Solve the following linear equation:

$\frac{(x-5)}{3}=\frac{(x-3)}{5}$

Q.

Solve the following equation and check your results: $x=\frac{4}{5}\left(x+10\right)$

Q. Let $\left[x\right]$ denote the greatest integer less than or equal to $x$. Then, the values of $x\in R$ satisfying the equation $[e^x{]}^2+[e^x+1]-3=0$ lies in the interval:

1. $[0,\frac{1}{e})$
2. $[1,e)$
3. $[0,{\log }_{e}2)$
4. $[{\log }_{e}2,{\log }_{e}3)$

Q.

Solve $(x-5)/2-(x-3)/5=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.

Q. A manufacturer has employed $5$ skilled men and $10$ semi-skilled men and makes two models $A$ and $B$ of an article. The making of one item of model $A$ requires $2$ hours work by a skilled men and $2$ hours work by a semi-skilled man. One item of model $B$ requires $1$ hour by a skilled man and $3$ hours by a semi-skilled man. No man is expected to work more than $8$ hours per day. The manufacturer's profit on an item of model $A$ is Rs.$15$ and on an item of model $B$ is Rs.$10.$ How many of items of each model should be made per day in order to produce maximize daily profit ? Formulate the above LPP and solve it graphically and find the maximum profit.

Q. Let $A=\{x\in R:x^2+(m-1)x-2(m+1)=0\}$ and $B=\{x\in R:(m-1)x^2+mx+1=0\},$ where $m\in R.$ If $A\cup B$ has exactly three distinct elements, then the number of possible values of $m$ is

Q.

What is the solution for $2\left(x+4\right)=12$?

Q.

Solve for $x$:

$\frac{3x-1}{5}-\frac{x}{7}=3$

Q. The function $f\left(x\right)=\left[x\right],$ where $\left[x\right]$ denotes greatest integer function, is continuous at

1. $4$
2. $-2$
3. $1$
4. $1.5$

Q. The shaded region in the figure is the solution set of the inequations.

1. $4x+5y\le 20,3x+10y\le 30,x\le 6,x,y\ge 0$
2. $4x+5y\le 20,3x+10y\le 30,x\ge 6,x,y\ge 0$
3. $4x+5y\ge 20,3x+10y\le 30,x\ge 6,x,y\ge 0$
4. $4x+5y\ge 20,3x+10y\le 30,x\le 6,x,y\ge 0$

Q. Two tailors, $A$ and $B$, earn Rs $300$ and Rs $400$ per day respectively. $A$ can stitch $6$ shirts and $4$ pairs of trousers while $B$ can stitch $10$ shirts and $4$ pairs of trousers perday. To find how many days should each of them work if it is desired to produce at least $60$ shirts and $32$ pairs of trousers at a minimum labour cost, formulate this as an LPP.

Q.

Solve the inequality $3x+2>-16,2x-3\le 11$.

1. $(-6,7]$

2. $[-6,7)$

3. $(-6,7)$

4. $[-6,7]$

Q. If $x=\sqrt{7}-\sqrt{5}$ and $y=\sqrt{13}-\sqrt{11}$, then

1. $x>y$
2. $x<y$
3. $x=y$
4. None of these

Q.

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 h of machine time and 3 h of machine time in its making while a cricket bat takes 3 hours of machine time and 1 h of craftsman's time. In a day, the factory has the availability of not more than 42 h of machine time and 24 h of craftsman's time.

(a) What number of rackets and bats must be made, if the factory is to work at full capacity?

(b) If the profits on rackets and on bats are Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.

Q. Find the intervals in which the function $f$ given by $f\left(x\right)=2x^3-3x^2-36x+7$ is

$\left(a\right)$ increasing
$\left(b\right)$ decreasing

Q. If $xϵR$ and $m=\frac{x^2}{(x^4-2x^2+4)}$, then m lies in the interval

1. $[0,\frac{1}{3}]$
2. $[0,\frac{1}{2}]$
3. $[0,\frac{1}{5}]$
4. $[0,\frac{1}{4}]$

Q.

Find the solution of the equation $2\mathrm{x}+5=14$.

Q.

$3x+4y=85$ and $6x+7y=20$. Find $x$ and $y$.

Q.

Solve the Equation and Verify the Answer :$5x-3=x+17$

Q. The solution set of $\frac{1}{x^2-9}\ge 0$ is

1. $(-3,3)$
2. $[-3,3]$
3. $(-\infty ,-3]\cup [3,\infty )$
4. $(-\infty ,-3)\cup (3,\infty )$

Q.

To promote making of toilets for women, an organization tried to generate awareness through (i) house calls (ii) letters, and (iii) announcements.

The cost for each mode per attempt is given below:

(i) $Rs.50$ (ii) $Rs.20$ (iii) $Rs.40$

The number of attempts made in three village $X$, $Y$ and $Z$ are given below:

(i)	(ii)	(iii)
X	400	300	100
Y	300	250	75
Z	500	400	150

Find the total cost incurred by the organization for three villages separately, using matrices.

Q. The feasible region of a LPP is shown in the figure. If $Z=5x+2y,$ then the maximum value of $Z$ occurs at

1. $(0,0)$
2. $(5,0)$
3. $(0,4)$
4. $(2,4)$

Q.

How to solve a system of inequalities without graphing $?$

Q.

In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in mg/tablet) are given as below

$\begin{array}{cccc}Tablets& Iron& Calcium& Vitamin\\ X& 6& 3& 2\\ Y& 2& 3& 4\end{array}$

The person needs atleast 18 mg of iron, 21 mg of calcium and 16 mg of vitamins. The price of each tablet of X and Y is Rs 2 and R1, respectively. How many tablets of each should the person take in order to stisfy the above requirement at the minimum cost?

Q. For the LPP; maximise $z=x+4y$ subject to the constraints $x+2y\le 2,x+2y\ge 8,x,y\ge 0$

1. $z_{\text{max}}=16$
2. Has no feasible solution
3. $z_{\text{max}}=4$
4. $z_{\text{max}}=8$

Q. For the LPP; maximize $z=5x-2y$ subject to the constraints $3x+2y\le 6,10x+9y\ge 45,x,y\ge 0$

1. $z_{min}=-15$
2. $z_{min}=0$
3. The LPP has no feasible solution.
4. $z_{min}=-6$

Q.

Solve the following linear equation:

$\frac{(3t-2)}{4}-\frac{(2t+3)}{3}=\frac{2}{3}-t$

Q.

If the product of three positive real numbers is 1 and their sum is greater than sum of their reciprocals, then

1. exactly one of them exceeds 1

2. each of them equals 1

3. exactly one of them is less than 1

4. None of these

Q.

Solve the following equation:

$\frac{2a}{3}-5=3$

Q.

Pick out the solution from the values given in the bracket next to each question. Show that the other values do not satisfy the equation:
$4m+5=21(1,5,4)$